Rabbi Yohanan said: A circular window needs 24 tefahim in its circumference. And two plus a bit from those need to be within ten [tefahim from the ground], so that if you make a square, some of it will be within ten.

This is difficult to understand.

Here's what the correct geometry would look like:

The circle has a diameter equal to the diagonal of the inscribed square, which in this is case 4√2, or four times the square root of two. The square root of two is an irrational number that begins 1.414213562..., and that Hazal generally approximate by 1+2/5. Next, the circumference of the circle is π (pi) times the diameter, or 4×π×√2. It's hard to do justice to π in a sentence, but it's an irrational number that begins 3.141592653... and is worth reading more about. Hazal generally approximate π by the round number 3, which has led to much discussion I won't get into here.

With Hazal's approximations for √2 and π, the window should have circumference 16+4/5 tefahim, and have about 4/5 tefahim of its height within ten tefahim of the ground. The precise values would be 4×π×√2 = 17.77153175... and 2√2−2 = 0.8284271247... respectively. Those aren't at all close to 24 and 2.

So how did Rabbi Yohanan get “24” and “2 and a bit”?

The Gemara concludes on Eruvin 76b that he got it from the judges of Caesarea, who stated what I will call “the Caesarea rule”:

Rashi understands the Caesarea rule to mean that for the perimeter of a square inside a circle, you take one third away from the circumference, or half of what will remain. So for a 4-by-4 square with perimeter 16, the circle that surrounds it would have circumference 24.

With the value of 3 for π, a circumference of 24 means our square has a diagonal of 8. The diagonal is twice the length of the side. "They consider all diagonals to be thus," says Rashi. But the diagonal of a square is obviously less than twice the length of the side. Just look at this diagram—is the red line not obviously much shorter than the blue line?

In fact, the Gemara on Sukkah 8a, which also quotes the Caesarea rule, notes that it is obviously wrong:

But we will bear with Rashi, who quite reasonably derived Rabbi Yohanan's “24” from the Caesarea rule.

What about the “2 and a bit”?

Rashi says that this length comes from the circumference, and it means 1 and a bit along the circle in each direction from its bottommost point.

This fits the words just as they sound, but is exceedingly difficult math-wise. Just look at the first diagram above and it's clear that you need the full bottom quarter of the circle to reach up to the square. If the circumference is 24, that arc would be 6. The correct value is in fact π√2 = 4.442882938... Rabbi Yohanan's 2 is way off.

The Mesivta on the daf cites a creative explanation of Rashi from the Leshon ha-Zahav to avoid this. He indeed gets Rashi's arc up to 6, but he has to stick quite a few critical words into Rashi that aren't there.

Other rishonim therefore explain this “2 and a bit” differently. Rivan in Tosafot, followed by the Ritva and Rashba, says that this is a vertical height from the bottom of the circle.

If the circle has a diameter of 8, as Rabbi Yohanan seems to think, and the square has a height of 4, that leaves 2 leftover above and below the square. With this explanation, Rabbi Yohanan successfully applies Rashi's Caesarea rule to the window. But he's still way off from reality. The vertical distance from the bottom of the circle to the bottom of the square should be 2√2−2 = 0.8284271247..., not 2.

Now, back to the size of our circle.

(2) Tosafot quote a yesh mefareshim that rescues the judges of Caesarea, but not Rabbi Yohanan. The Rashba and Ritva also quote this explanation from Teshuvat ha-Ge'onim. They say that the Caesarea rule describes area, not perimeter. The picture is clearest like this:

If we assume that π is 3, then the area of circle is indeed one quarter less than the outer circle, or π/4 of it. And the inner square is then half the area of the outer square, which you can see clearly by breaking the squares into right-angled triangles, as I did above with a light dashed line.

In terms of the wording, it's a bit awkward to read "half" as referring to the outer square, when the nearest antecedent is the circle. But on the other hand, the fractions are consistently "inside" fractions, whereas for Rashi the Caesarean quarter is an "inside" fraction and the Caesarean half is an "outside" fraction. On the whole, I'd say this explanation for the Caesarea rule is much more satisfying.

But Tosafot note a major drawback. Our Gemara, both in Eruvin and in Sukkah, understand the Caesarea rule as talking about perimeters, not areas. The amora'im had to completely misunderstand the rule.

(An aside about the above diagram: The Vilna Shas prints a diagram in Tosafot like mine, but with the inside square standing with sides parallel to the outside square, not at a diagonal as I have it. With my way it's easier to see how the area is half. The Maharsha thus says the diagram should look like mine. But the Maharshal there endorses the Vilna Shas version. The Mesivta in Yalkut Bi'urim describes this as a dispute, which is a little funny, given that the square has the same area any way you turn it. I'll give them all the benefit of the doubt and assume they were arguing over pedagogy, not geometry.)

(3) Next, the Rashba quotes the explanation of the Ra'avad, who says that the Caesarea rule is talking about lengths, but rounding up as a stringency:

The problem is that when you simply say "circle," I don't imagine this fancy shape. And the early stages of the Gemara clearly assumed a normal circle.

(Another little point about diagrams: When the Steinsaltz Talmud presents the shape described by Ritva and Rashba, the artist gets it wrong. He doesn't show a full semicircle for each arc, but rather something like a third of a circle for each arc.)

Rabbi Yohanan said “24” about the circle's area, not its circumference. He gets this by getting an area of 32 for the outer square, then subtracting a quarter (i.e., taking π/4) to get 24 for the circle. The “2 and a bit” also refers to the area of the circle under the 4-by-4 square.

The Meiri writes about two thousand words interpreting this Gemara. Here is where he explains Rabbi Yohanan:

This works great for the math. Rabbi Yohanan applied the Caesarea rule properly—better, in fact, than most rishonim were able to apply it. And the “2 and a bit” is consistent with the “24” by also referring to area.

There are just two things I don't like in terms of wording. Rabbi Yohanan's phrase be-heikefo esrim ve-arba tefahim sounds more like a length than an area. The Gemara also assumes throughout its discussion that Rabbi Yohanan was using lengths, and then assumes that we understand the leap to areas when they quote the Caesarea rule. But neither of these problems are so bad, I think.

The Meiri cites his explanation from gedolei ha-rishonim. Footnote 3 on page 293 of the Hirshler edition notes that the Eisenstein manuscript (I sound so scholarly!) has a few extra lines identifying those rishonim as the Ba'al ha-Ma'or and R' Yehudah ibn Tibbon. He describes how they reached their explanation:

Rabbi Yohanan said “24” not about the circle, but of a square that surrounds this circle. The outer square would have a perimeter of 4×4×√2, equal to 22+2/5 if we take √2 as 1+2/5. Then 24 is his approximation for 22+2/5.

But the Gemara explicitly understands “24” as the circumference of the window itself, not a theoretical square. The Vilna Gaon would have to say either that the setama of the Gemara misunderstood Rabbi Yohanan, or that this assumption changed when the Gemara quoted the Caesarea rule. Combine that with the roughness of the approximation, and this explanation doesn't seem very satisfying.

(7) The Hagahot ha-Bah, on the Rif, via Steinsaltz, uses the shape shown below, but I'm not sure how it translates into a physical window.

(8) One last suggestion: The periodical Tehumin, in volume 19 (5759), page 456, has an article by an architect named Zvi Spalter entitled “ha-π ha-Hazali.” Spalter presents some apologetics for all the places where Hazal approximate π as 3. His original arguments are dohak, in my humble opinion, and he loses credibility in supremely ironic fashion with his second sentence. He writes that π is equal to 22/7, which is equal to 3.14159. In fact, 22/7 = 3.124857124857..., and π is not a rational number.

Spalter's suggestion for Eruvin 76b is that Rabbi Yohanan considered a window in a wall a couple tefahim thick, and that the window was wider on the inside of the wall than the outside of the wall. The 4-by-4 square is measured at the outside wall, and the 24-around circle is measured at the inside of the wall. He says that using the word "window" as opposed to "opening" or "hole" is evidence for this. I guess he realized that such a leap requires evidence.

This idea actually crossed my mind too, but I dismissed it pretty quickly. The width of the wall is a critical detail for Rabbi Yohanan to leave out, and for none of the mefareshim to even consider. And "window" doesn't at all suggest a third dimension here, considering that most of the daf talks about windows and stays in two dimensions.

·

That makes eight different interpretations for what Rabbi Yohanan meant. Was he talking about edge lengths, areas, or heights? What did the judges of Caesarea mean, and did Rabbi Yohanan understand them properly? Each interpretation takes a different direction, weighing the needs to fit the words of the text and be accurate with the math.

I am pretty much persuaded by the Meiri and his cohort, who say that Rabbi Yohanan correctly applied a true (with π as 3) Caesarean principle about areas. The thing that bothers me most about his interpretation is the flow of the Gemara.

My suggestion, then, is to combine the Meiri with Rashi. The Meiri found the correct explanation of what both the judges of Caesarea and Rabbi Yohanan were saying. But Rashi correctly explained the whole sugya here as understanding both statements as discussing lengths, not areas. The anonymous amora who composed it misunderstood Rabbi Yohanan and the judges of Caesarea.

This is a similar approach to Tosafot, except that I'm also saving Rabbi Yohanan, and instead pinning the mistake on an anonymous amora. So sincere apologies, anonymous amora,for putting the blame on you, but if I'm right, then you confused most mefareshim until today about what these statements meant. If I'm wrong, please forgive me.

Now, if you want to see squares merging with circles, the real merging with the imaginary, and existence merging with non-existence—all in one short, simple, beautiful equation—go read about Euler's identity.

1 comment:

Great blog post thank you! Can you interpret this passage? Does it bear any relation?

“What is the meaning of שֶׁבֶט (shevet), staff? It is something simple/whole and not square. What is the reason? Because it is impossible to have one square [i.e., a tribe in battle formation?] inside another square. A circle inside a square can move. A square inside a square cannot move” (Bahir §114).